The prerequisite is deep mathematical, rational and process-oriented thinking. You can get that lots of ways, but an MBA alone won't give it to you.
I disagree with the notion that the path through management should be through demonstrating your value as an engineer. If a high performing engineer desires a move into management, I think that should certainly be considered, but I have no evidence that top engineers transition into being top managers.
Being a successful engineer is evidence that you understand processes and systems very deeply. What remains is interpersonal skill and basic economics. Being a successful business person is evidence of the complement.
Couple this with the observation that it is easier to find people who picked up economics and interpersonal skills without training than it is to find people with deep untrained technical skill, because interactions with other people and decisions about money occur every day so any motivated somebody with a curious disposition may well have been pondering and improving those skills their whole life.
Remember, it's not that every engineer you find will be better rounded, it's just that those magical omnibus people are more likely to be found among the engineering ranks than the business ranks, especially when engineering jobs give a much more reliable salary out of college so that anyone with engineering skill and business skill is incentivized to study the former in school.
The problem with your path to management is that many engineers have no desire to manage, and anecdotally, those who do, end up being ill-suited for management.
The best managers I've had are not top engineers, rather they're managers that are willing to understand and listen to their top performing engineers.
Google/Microsoft/etc. offer PM positions to students directly out of undergrad. From my experience, the skillset between a top-performing engineer is vastly different from that of a top-performing and well liked PM.